Bounds for one-dimensional rings of finite cohen-macaulay type

Abstract

Wiegand, R. and S. Wiegand, Bounds for one-dimensional rings of finite Cohen-Macaulay type, Journal of Pure and Applied Algebra 93 (1994) 311-342. Let R be an integral domain finitely generated as an algebra over a field of characteristic not equal to 2 (or the localization of such a ring at some multiplicatively closed set); and assume that, for each maximal ideal ▪, there is a bound on the ranks of the indecomposable finitely generated torsion-free R ▪ -modules. We show that the only possible ranks for such indecomposable modules over R are 1, 2, 3, 4, 5, 6, 8, 9 and 12. An example having indecomposables of each of these ranks is constructed over the field of rational numbers. Furthermore, over a broader class of reduced one-dimensional rings, the only possible ranks for indecomposable finitely generated torsion-free modules of constant rank are also 1, 2, 3, 4, 5, 6, 8, 9 and 12.